Look into credit unions for which you might be eligible, as well as smaller regional banks that don't deal in exotic financial products and in which the management team still feels a personal responsibility to their depositors. patio11's recent blog post on Japan has a good section on what that's like:
http://www.kalzumeus.com/2014/11/07/doing-business-in-japan/ (ctrl-f for "Taro", read about his banking relationship manager)
There are still banks in the US that operate like that in principle, if not in degree - no exotic products, strict adherence to traditional corporate lending standards, extensive knowledge of their customers, and a culture ingrained first and foremost with personal responsibility to the soundness of the local economy.
Best example I know of is First Hawaiian Bank [1], having worked there a few years. It is actually the longest continually-running bank in the US, having been founded in 1858 and survived every major depression and financial crisis since. They're not on the bleeding edge of tech or quantitative finance, but that's one of the things that saved them from being a victim of the financial crisis (aka, the Battlestar Galactica of banking).
Other banks like that scattered around the country too. You can start some research with the various bank rating sites:
http://www.bankrate.com/rates/safe-sound/bank-ratings-search...
http://www.bauerfinancial.com/
[1]:https://www.fhb.com/en/inside-fhb/our-rich-history/